Integration by Parts

In this page integration by parts we are going to see where we have to apply this method. Integration-by-parts is one of the method basically used o find the integral when the integrand is a product of two different kind of function.

Formula :

∫ u dv = uv - ∫ v du

Given Integral

∫ log x dx

∫ tan⁻
¹ x dx

∫ xⁿ log x dx

∫ xⁿ tan⁻
¹ x dx

∫ xⁿ e^(ax) dx

∫ xⁿ sin xdx

∫ xⁿ Cos xdx

U

log x

tan⁻
¹ x

log x

tan⁻
¹ x

xⁿ

xⁿ

xⁿ

dv

dx

dx

xⁿ dx

xⁿ dx

e^(ax) dx

sin xdx

Cos xdx

Now let us see some example problems to understand this topic better.

Example 1:

Integrate x sec² x with respect to x

Solution:

In this problem we must apply the concept integrating-by-parts. Because here we have two function which are multiplying. To integrate this we must apply this concept.

∫ x sec² x dx

u = x dv = sec² x dx

du = dx ∫ dv = ∫ sec² x dx

v = tan x

Formula :

∫ u dv = uv - ∫ v du

∫ x sec² x dx = x (tan x) - ∫ (tan x) dx

= x (tan x) - sec² x + C

Example 2:

Integrate Sin⁻¹ x with respect to x

Solution:

In
this problem we must apply the concept integrating-by-parts. Because
here we have two function which are multiplying. To integrate this we
must apply this concept.